Origin of Monte Carlo Methods

1.6.1 Origin of Monte Carlo Methods

In most of the applications of probability theory one makes a mathematical formulation of

a stochastic problem (i.e., a problem where chance plays some part) and then solves the problem by using analytical or numerical methods. In the Monte Carlo method one does the opposite; a mathematical or physical problem is given, and one constructs a numerical game of chance , the mathematical analysis of which leads to the same equations as the given problem for, e.g., the probability of some event, or for the mean of some random variable in the game. One plays it N times and estimates the relevant quantities by traditional statistical methods. Here N is a large number, because the standard deviation of a statistical estimate √ typically decreases only inversely proportionally to N .

1.6. Monte Carlo Methods

65 The idea behind the Monte Carlo method was used by the Italian physicist Enrico Fermi

to study neutron diffusion in the early 1930s. Fermi used a small mechanical adding machine for this purpose. With the development of computers larger problems could be tackled. At Los Alamos in the late 1940s the use of the method was pioneered by von Neumann, 21

Ulam, 22 and others for many problems in mathematical physics including approximating complicated multidimensional integrals. The picturesque name of the method was coined by Nicholas Metropolis.

The Monte Carlo method is now so popular that the definition is too narrow. For instance, in many of the problems where the Monte Carlo method is successful, there is already an element of chance in the system or process which one wants to study. Thus such games of chance can be considered numerical simulations of the most important aspects. In this wider sense the “Monte Carlo methods” also include techniques used by statisticians since around 1900, under names like experimental or artificial sampling. For example, statistical experiments were used to check the adequacy of certain theoretical probability laws that had been derived mathematically by the eminent scientist W. S. Gosset. (He used the pseudonym “Student” when he wrote on probability.)

Monte Carlo methods may be used when the changes in the system are described with

a much more complicated type of equation than a system of ordinary differential equations. Note that there are many ways to combine analytical methods and Monte Carlo methods. An important rule is that if a part of a problem can be treated with analytical or traditional

numerical methods, then one should use such methods.

The following are some areas where the Monte Carlo method has been applied: (a) Problems in reactor physics; for example, a neutron, because it collides with other

particles, is forced to make a random journey. In infrequent but important cases the neutron can go through a layer of (say) shielding material (see Figure 1.6.1).

(b) Technical problems concerning traffic (in telecommunication systems and railway networks; in the regulation of traffic lights, and in other problems concerning auto- mobile traffic).

(c) Queuing problems. (d) Models of conflict.

(e) Approximate computation of multiple integrals. (f) Stochastic models in financial mathematics.

Monte Carlo methods are often used for the evaluation of high-dimensional (10 to 100) integrals over complicated regions. Such integrals occur in such diverse areas as

21 John von Neumann was born János Neumann in Budapest 1903, and died in Washington D.C. 1957. He studied under Hilbert in Göttingen in 1926–27, was appointed professor at Princeton University in 1931, and in

1933 joined the newly founded Institute for Advanced Studies in Princeton. He built a framework for quantum mechanics, worked in game theory, and was one of the pioneers of computer science.

22 Stanislaw Marcin Ulam, born in Lemberg, Poland (now Lwow, Ukraine) 1909, and died in Santa Fe, New Mexico, USA, 1984. Ulam obtained his Ph.D. in 1933 from the Polytechnic institute of Lwow, where he studied

under Banach. He was invited to Harvard University by G. D. Birkhoff in 1935 and left Poland permanently in 1939. In 1943 he was asked by von Neumann to come to Los Alamos, where he remained until 1965.

66 Chapter 1. Principles of Numerical Calculations

Figure 1.6.1. Neutron scattering.

quantum physics and mathematical finance. The integrand is then evaluated at random points uniformly distributed in the region of integration. The arithmetic mean of these function values is then used to approximate the integral; see Sec. 5.4.5.

In a simulation, one can study the result of various actions more cheaply, more quickly, and with less risk of organizational problems than if one were to take the corresponding actions on the actual system. In particular, for problems in applied operations research, it is quite common to take a shortcut from the actual system to a computer program for the game of chance, without formulating any mathematical equations. The game is then a model of the system. In order for the term “Monte Carlo method” to be correctly applied, however, random choices should occur in the calculations. This is achieved by using so-called random numbers ; the values of certain variables are determined by a process comparable to dice throwing. Simulation is so important that several special programming languages have been developed exclusively for its use. 23